The group of operators on
![](http://upload.wikimedia.org/math/f/9/3/f930fe54c701b3ddd5eb7510a6a1cad9.png)
generated by a single translation
![](http://upload.wikimedia.org/math/4/1/a/41a9b9a01bb4dc28e69d56d9bcac9317.png)
and a single modulation
![](http://upload.wikimedia.org/math/5/3/d/53d3881fdc61ba3d45fbc4b030ef1356.png)
is a familiar object of study in signal analysis. It forms a unitary representation of the so-called discrete Heisenberg group. As such, its analysis yields easily to classical methods when
![](http://upload.wikimedia.org/math/4/d/1/4d1b7b74aba3cfabd624e898d86b4602.png)
is rational. But when
![](http://upload.wikimedia.org/math/4/d/1/4d1b7b74aba3cfabd624e898d86b4602.png)
is irrational, it provides instructive concrete examples of several pathological representation-theoretic phenomena. We shall introduce the relevant concepts from representation theory, discuss the rational case briefly, and then explore the irrational case in more detail.